Multiple choice

In a bag there are $N$ balls out of which $R$ are red.If $n$ balls are drawn at random from the bag, the probability that $r$ of them are red is

  1. $\displaystyle \frac{n_{C_{r}}}{^NC_{n}}$
  2. $\displaystyle \frac{R_{C_{r}}}{^NC_{n}}$
  3. $\displaystyle \frac{^{R}C_{r}\times ^{N-R}c_{n-r}}{^{N}C_{n}}$
  4. $\displaystyle \frac{^{N}C_{r}}{^{R}C_{n}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

This is a classic hypergeometric distribution problem. The number of ways to choose r red balls from R is RCr, and the number of ways to choose the remaining (n-r) balls from (N-R) is (N-R)C(n-r). The total ways to choose n balls from N is NCn.

AI explanation

This scenario describes the hypergeometric distribution, used to find the probability of a specific number of successes in draws without replacement. The total ways to draw n balls from the bag is given by the combination N choose n. The favourable ways to choose exactly r red balls from the available R red balls is R choose r, and the remaining balls are chosen from the non-red balls, yielding N minus R choose n minus r. The probability formula combines these terms as (R choose r multiplied by N minus R choose n minus r) divided by N choose n.

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